%0 Journal Article
%A Estaji, A. A.
%A Karimi Feizabadi, A.
%A Robat Sarpoushi, M.
%T Sums of Strongly z-Ideals and Prime Ideals in ${mathcal{R}} L$
%J Iranian Journal of Mathematical Sciences and Informatics
%V 15
%N 1
%U http://ijmsi.ir/article-1-1025-en.html
%R 10.29252/ijmsi.15.1.23
%D 2020
%K Frame, Ring of real-valued continuous functions, z-Ideal, Strongly z-ideal.,
%X It is well-known that the sum of two $z$-ideals in $C(X)$ is either $C(X)$ or a $z$-ideal. The main aim of this paper is to study the sum of strongly $z$-ideals in ${mathcal{R}} L$, the ring of real-valued continuous functions on a frame $L$. For every ideal $I$ in ${mathcal{R}} L$, we introduce the biggest strongly $z$-ideal included in $I$ and the smallest strongly $z$-ideal containing $I$, denoted by $I^{sz}$ and $I_{sz}$, respectively. We study some properties of $I^{sz}$ and $I_{sz}$. Also, it is observed that the sum of any family of minimal prime ideals in the ring ${mathcal{R}} L$ is either ${mathcal{R}} L$ or a prime strongly $z$-ideal in ${mathcal{R}} L$. In particular, we show that the sum of two prime ideals in ${mathcal{R}} L$ such that are not a chain, is a prime strongly $z$-ideal.
%> http://ijmsi.ir/article-1-1025-en.pdf
%P 23-34
%& 23
%!
%9 Research paper
%L A-10-2462-1
%+ Hakim Sabzevari University
%G eng
%@ 1735-4463
%[ 2020